Network Security - asymmetric key encryption

1. Asymmetric Key Encryption

Definition:
Asymmetric encryption uses two keys:

• Public key: Known to everyone, used for encryption

• Private key: Kept secret, used for decryption

Key points:

• Eliminates the need to share secret keys over insecure channels.

• Slower than symmetric encryption, often used to exchange symmetric keys.

• Enables digital signatures for authentication and integrity.

2. RSA (Rivest–Shamir–Adleman)

• Developed: 1977

• Type: Public-key cryptosystem

• Based on: Difficulty of factoring large prime numbers

How it works (simplified):

1. Generate two large prime numbers pp and qq

2. Compute n=p×qn = p \times q (used as modulus)

3. Calculate ϕ(n)=(p−1)(q−1)\phi(n) = (p-1)(q-1)

4. Choose a public key exponent ee (1 < e < φ(n), gcd(e, φ(n)) = 1)

5. Compute private key dd such that d×e≡1 (mod ϕ(n))d \times e \equiv 1 \ (\text{mod } \phi(n))

Encryption: C=Memod  nC = M^e \mod n
Decryption: M=Cdmod  nM = C^d \mod n

Pros:

• Well-studied and widely used

• Supports both encryption and digital signatures

Cons:

• Requires large key sizes (2048+ bits) for strong security

• Slower than symmetric algorithms

3. ECC (Elliptic Curve Cryptography)

• Developed: 1985 (by Victor Miller and Neal Koblitz)

• Type: Public-key cryptosystem

• Based on: Elliptic curve discrete logarithm problem (ECDLP)

How it works (simplified):

1. Choose an elliptic curve y2=x3+ax+by^2 = x^3 + ax + b over a finite field

2. Select a base point GG on the curve

3. Private key dd is a random number

4. Public key Q=d⋅GQ = d \cdot G (scalar multiplication on the curve)

Encryption and signature:

• Uses operations on points of the elliptic curve

• Provides same security as RSA with much smaller key sizes

Pros:

• Smaller keys → faster computation and less storage

• Efficient for mobile and IoT devices

• Strong security with 256-bit key comparable to 3072-bit RSA

Cons:

• More complex mathematics than RSA

• Patent issues in early years (mostly expired now)

Comparison: RSA vs ECC